Quantum Heegaard diagrams and knot Floer Homology
Cristina Ana-Maria Anghel, Andr\'as Juh\'asz

TL;DR
This paper introduces quantum Heegaard diagrams that unify the Alexander and Jones polynomials through a geometric intersection model, providing new insights into knot invariants and their categorifications.
Contribution
It constructs quantum Heegaard diagrams from braid presentations, linking knot Floer homology with a new geometric framework that recovers both Alexander and Jones polynomials.
Findings
Quantum Heegaard diagrams unify Alexander and Jones polynomials.
Knot Floer homology is viewed as a categorification of the Alexander polynomial.
A new geometric intersection formula for the Jones polynomial is proposed.
Abstract
Given a knot presented as a braid closure, we construct a unified intersection model for the Alexander and Jones polynomials of the knot via what we call quantum Heegaard diagrams. These diagrams are obtained by stabilising the disc model of the first author, which we show are doubly-pointed Heegaard diagrams of the knot together with an additional set of base points. We identify the Alexander grading in the disc model with the Alexander grading in the Heegaard diagram. As the Lagrangian intersection Floer homology of the Heegaard tori in the symmetric power of the Heegaard surface is knot Floer homology, we can view knot Floer homology as a natural categorification of the Alexander polynomial arising from the disc model. The additional base points let us define a new grading on the intersection between the Heegaard tori, which we call quantum Alexander grading. Combining this with…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology
